How to change shape keys with the same name in different objects at the same time
Yes, you can do it without a rig, but those shape_keys are on different objects and shape_keys from different objects can not be mixed, only transferred if the objects have the same vertex count. Choice one, joining objects:
So considering the pure shape_key blend/mix way, the function is found right to the shape_key list when you click on the "v" button.
If you join the objects, i would suggest to rename the shape_keys to reflect which part they belong to (face, eyebrows) beforehand, as that combines the shape_keys from both objects in one list. That allows then to adjust all keys and you can set face_angry_left and eyebrow_angry_left (just an example for the names) to 1.0, after which you click on that mentioned "v" button and select "+ New Shape From Mix". That will create a new key with the effect of both previous set to 1 keys, means you get a shape_key adjusting both values at the same time.
In the picture you see the button mentioned and the functions.



Choice 2: Using a driver, which you can set up as a new shape_key on your head for example or a custom property value under your head. It does not really matter, as long as you have a value that you right click on, and choose "Copy As New Driver" function (see below, in second picture right click on any value really you get the second last option in the popup menu). After that you go in your eyebrows and select the shape_key for angry_left for example and right click over the value to then select the "Paste Driver" function. That will color the value purple and connects the previously used value (New Driver) to the now colored value. You can do the same with the face where you select the angry_left value of the shape_keys and right click once again to also "Paste Driver" here. Well, you now have both objects essentially remotecontrolled with the "New Driver", only problem with this solution is that you can't change the values separately anymore. To work around that you could copy the shape_keys before pasting the driver on them, but if you ask me, i would consider to consider choice 1.
To not leave it out, copying a shape_key goes by setting only the value of the to be copied key to 1 and then click on the "v" button and use the "+ New Shape From Mix" function, generating a clone.